If $\tan x$ is an integrating factor of the differential equation $\frac{dy}{dx} + Py = Q$, then $P$ can be

The solution of the differential equation $\frac{dy}{dx} + x \sin^2 y = \sin y \cos y$ is:

The integrating factor of the differential equation $(1 + t^2) + (x - e^{\tan^{-1} t}) \frac{dt}{dx} = 0$ is

Find the general solution of the differential equation: $\frac{dy}{dx} + 2y = \sin x$.

$A$ function $y = f(x)$ satisfies $(x + 1)f'(x) - 2(x^2 + x)f(x) = \frac{e^{x^2}}{(x + 1)}$. If $f(0) = 5$,then $f(x)$ is:

Let $y=y(x)$ be the solution of the differential equation $(\tan x)^{1/2} dy = (\sec^3 x - (\tan x)^{3/2} y) dx$, where $0 < x < \frac{\pi}{2}$ and $y(\frac{\pi}{4}) = \frac{6\sqrt{2}}{5}$. If $y(\frac{\pi}{3}) = \frac{4}{5}\alpha$, then $\alpha^4$ equals . . . . . . .